I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, such as locally finite covers and subordinate partitions of unity, a tubular neighborhood theorem, smoothing of submanifolds...

The main difference from the standard results is that I require uniformly bounded estimates, so for example the tubular neighborhood must have a uniformly finite size and a uniformly bounded diffeomorphism. This means that I cannot simply generalize the standard proofs.

I don't have any specific references for partitions of unity or tubular neighborhoods, but I can suggest two authors who have written a lot about bounded geometry manifolds and metric spaces: John Roe and Guoliang Yu. You might try "Lectures on Coarse Geometry" by Roe, for example; even if your specific questions are not answered there, I bet it would be fruitful to chase down the references.
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Paul SiegelJun 1 '11 at 14:35

I recommend the books of Jost and Gallot-Hulin-Lafontaine, as well as Gromov's classic "Metric Structures for Riemannian and Non-Riemannian Spaces". For tubular neighborhood, see the classic paper of Heintze-Karcher. For locally finite covers and partitions of unity, check the papers of Stefan Peters and Greene-Wu on convergence of Riemannian manifolds and the references cited by them.
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Deane YangJun 6 '11 at 14:20

Thanks, I finally got a hold of the book by Eichhorn, and there are indeed many results and references in there! I haven't found the uniform tubular neighborhood and smoothed manifold theorems yet, so I guess these might be new then.
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Jaap ElderingJun 7 '13 at 14:19